INTERNATIONAL GONGRESS $0\; 19\; 333\; 5Q328\; 0\; 317\; 0Q314\; 0\; 280\; 1T180\; 2Q118\; 2\; 85\; 2T49\; 1Q31\; 1\; 31\; 11Q31\; 13\; 34\; 25Q38\; 41\; 42\; 43T65\; 46Q92\; 46\; 125\; 49Q139\; 52\; 144\; 61Q147\; 65\; 216\; 339T285\; 628Z">$ |{\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]|$$ bounded by (2.5) for all $\ell$$\mathrm{ℓ}$ℓ\ell$\mathrm{ℓ}$; and infinitely many totally real degree $n{S}_n$$n{S}_n$nS_(n)n S_{n}$n{S}_n$-fields $K$$K$KK$K$ with ${\mathrm{C}\mathrm{l}}_K$${\mathrm{C}\mathrm{l}}_K$Cl_(K)\mathrm{Cl}_{K}${\mathrm{C}\mathrm{l}}_K$ containing an element of exact order $\ell$$\mathrm{ℓ}$ℓ\ell$\mathrm{ℓ}$ and $|{\mathrm{C}\mathrm{l}}_K[\ell ]|$$\left|{\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]\right|$|Cl_(K)[ℓ]|\left|\mathrm{Cl}_{K}[\ell]\right|$|{\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]|$ bounded by (2.5).) What happens when $G$$G$GG$G$ does not have a unique minimal nontrivial normal subgroup? Here is an open question: in general, when $N$$N$NN$N$ is a nontrivial normal subgroup of $G$$G$GG$G$ (not necessarily unique or minimal), what is the true order of growth of (4.12) as $X\to \mathrm{\infty }$$X→\mathrm{∞}$X rarr ooX \rightarrow \infty$X→\mathrm{∞}$ ? Questions about this "intersection multiplicity" are gathered in [62].

Third, increased attention has turned to bounding $\ell$$\mathrm{ℓ}$ℓ\ell$\mathrm{ℓ}$-torsion in class groups for all fields in special families specified by the Galois group: that is, proving property ${\mathbf{C}}_{\mathcal{F},\ell }(\mathrm{\Delta })$${\mathbf{C}}_{\mathcal{F},\mathrm{ℓ}}(\mathrm{Δ})$C_(F,ℓ)(Delta)\mathbf{C}_{\mathscr{F}, \ell}(\Delta)${\mathbf{C}}_{\mathcal{F},\mathrm{ℓ}}(\mathrm{Δ})$ for some $\mathrm{\Delta }<1/2$$\mathrm{Δ}<1/2$Delta < 1//2\Delta<1 / 2$\mathrm{Δ}<1/2$. First, Klüners and Wang have proved ${\mathbf{C}}_{\mathcal{F},p}(0)$${\mathbf{C}}_{\mathcal{F},p}(0)$C_(F,p)(0)\mathbf{C}_{\mathscr{F}, p}(0)${\mathbf{C}}_{\mathcal{F},p}(0)$ for the family ${\mathcal{F}}_{p^r}(G;X)$${\mathcal{F}}_{p^r}(G;X)$F_(p^(r))(G;X)\mathscr{F}_{p^{r}}(G ; X)${\mathcal{F}}_{p^r}(G;X)$ for any $p$$p$pp$p$-group $G$$G$GG$G$; this generalizes the application of genus theory to prove ${\mathbf{C}}_{2,2}(0)$${\mathbf{C}}_{2,2}(0)$C_(2,2)(0)\mathbf{C}_{2,2}(0)${\mathbf{C}}_{2,2}(0)$ [54].

Second, let $G=(\mathbb{Z}/p\mathbb{Z}{)}^r$$G=(\mathbb{Z}/p\mathbb{Z}{)}^r$G=(Z//pZ)^(r)G=(\mathbb{Z} / p \mathbb{Z})^{r}$G=(\mathbb{Z}/p\mathbb{Z}{)}^r$ be an elementary abelian group of rank $r\ge 2$$râ‰\yen 2$r >= 2r \geq 2$râ‰\yen 2$, with $p$$p$pp$p$ prime. Wang has shown that for every $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$, within the family of Galois $G$$G$GG$G$-fields $K/\mathbb{Q}$$K/\mathbb{Q}$K//QK / \mathbb{Q}$K/\mathbb{Q}$, property ${\mathbf{C}}_{\mathcal{F},\ell }(1/2-\delta (\ell ,p))$${\mathbf{C}}_{\mathcal{F},\mathrm{â„“}}(1/2−\mathrm{δ}(\mathrm{â„“},p))$C_(F,â„“)(1//2-delta(â„“,p))\mathbf{C}_{\mathscr{F}, \ell}(1 / 2-\delta(\ell, p))${\mathbf{C}}_{\mathcal{F},\mathrm{â„“}}(1/2−\mathrm{δ}(\mathrm{â„“},p))$ holds for some $\delta (\ell ,p)>0$$\mathrm{δ}(\mathrm{â„“},p)>0$delta(â„“,p) > 0\delta(\ell, p)>0$\mathrm{δ}(\mathrm{â„“},p)>0$ [91]. Since the savings $\delta (\ell ,p)$$\mathrm{δ}(\mathrm{â„“},p)$delta(â„“,p)\delta(\ell, p)$\mathrm{δ}(\mathrm{â„“},p)$ is independent of the rank, for $r$$r$rr$r$ sufficiently large this is better than ${\mathbf{C}}_{\mathcal{F},\ell }\left({\mathrm{\Delta }}_{\mathrm{G}\mathrm{R}\mathrm{H}}\right)$${\mathbf{C}}_{\mathcal{F},\mathrm{â„“}}\left({\mathrm{Δ}}_{\mathrm{G}\mathrm{R}\mathrm{H}}\right)$C_(F,â„“)(Delta_(GRH))\mathbf{C}_{\mathscr{F}, \ell}\left(\Delta_{\mathrm{GRH}}\right)${\mathbf{C}}_{\mathcal{F},\mathrm{â„“}}\left({\mathrm{Δ}}_{\mathrm{G}\mathrm{R}\mathrm{H}}\right)$. The method of proof plays off the interaction of three facts arising from the precise structure of $G$$G$GG$G$ : first, $|{\mathrm{C}\mathrm{l}}_K[\ell ]|$$\left|{\mathrm{C}\mathrm{l}}_K[\mathrm{â„“}]\right|$|Cl_(K)[â„“]|\left|\mathrm{Cl}_{K}[\ell]\right|$|{\mathrm{C}\mathrm{l}}_K[\mathrm{â„“}]|$ factors as a product of $|{\mathrm{C}\mathrm{l}}_F[\ell ]|$$\left|{\mathrm{C}\mathrm{l}}_F[\mathrm{â„“}]\right|$|Cl_(F)[â„“]|\left|\mathrm{Cl}_{F}[\ell]\right|$|{\mathrm{C}\mathrm{l}}_F[\mathrm{â„“}]|$ where $F$$F$FF$F$ varies over the $\approx p^{r-1}$$≈p^{r−1}$~~p^(r-1)\approx p^{r-1}$≈p^{r−1}$ many degree $p$$p$pp$p$ subfields of $K$$K$KK$K$, so it suffices to bound one of these factors nontrivially. Second, any rational prime splits completely in $\approx p^{r-2}$$≈p^{r−2}$~~p^(r-2)\approx p^{r-2}$≈p^{r−2}$ of these subfields, so at least one subfield has a positive proportion of primes splitting completely in it. Third, the sizes of the discriminants of the subfields can be played against each other, so that known prime-counting results (which may a priori seem to count primes that are "too large") suffice for the application of the Ellenberg-Venkatesh criterion. This is an interesting counterpoint to the methods described earlier. In another direction, Wang has developed the notion of a forcing extension; certain nilpotent groups can be built from elementary $p$$p$pp$p$-groups via forcing extensions. If $G^{\mathrm{\prime }}$$G^{\mathrm{′}}$G^(')G^{\prime}$G^{\mathrm{′}}$ is constructed from $G$$G$GG$G$ by a forcing extension, then ${\mathbf{C}}_{{\mathcal{F}}^{\mathrm{\prime }},\ell }\left({\mathrm{\Delta }}^{\mathrm{\prime }}\right)$${\mathbf{C}}_{{\mathcal{F}}^{\mathrm{′}},\mathrm{â„“}}\left({\mathrm{Δ}}^{\mathrm{′}}\right)$C_(F^('),â„“)(Delta^('))\mathbf{C}_{\mathscr{F}^{\prime}, \ell}\left(\Delta^{\prime}\right)${\mathbf{C}}_{{\mathcal{F}}^{\mathrm{′}},\mathrm{â„“}}\left({\mathrm{Δ}}^{\mathrm{′}}\right)$ can be deduced from ${\mathbf{C}}_{\mathcal{F},\ell }(\mathrm{\Delta })$${\mathbf{C}}_{\mathcal{F},\mathrm{â„“}}(\mathrm{Δ})$C_(F,â„“)(Delta)\mathbf{C}_{\mathscr{F}, \ell}(\Delta)${\mathbf{C}}_{\mathcal{F},\mathrm{â„“}}(\mathrm{Δ})$, for some $\mathrm{\Delta },{\mathrm{\Delta }}^{\mathrm{\prime }}<1/2$$\mathrm{Δ},{\mathrm{Δ}}^{\mathrm{′}}<1/2$Delta,Delta^(') < 1//2\Delta, \Delta^{\prime}<1 / 2$\mathrm{Δ},{\mathrm{Δ}}^{\mathrm{′}}<1/2$, where $\mathcal{F}$$\mathcal{F}$F\mathscr{F}$\mathcal{F}$ is the family of $G$$G$GG$G$-extensions and ${\mathcal{F}}^{\mathrm{\prime }}$${\mathcal{F}}^{\mathrm{′}}$F^(')\mathscr{F}^{\prime}${\mathcal{F}}^{\mathrm{′}}$ is the family of $G^{\mathrm{\prime }}$$G^{\mathrm{′}}$G^(')G^{\prime}$G^{\mathrm{′}}$-extensions [89].

All of the results mentioned in this section (except where genus theory suffices) directly apply or build on the Ellenberg-Venkatesh criterion. Can this criterion be strengthened? Ellenberg has suggested some possible improvements in [32]. In particular, let $\eta (K):=inf\{H_K(\alpha ):K=\mathbb{Q}(\alpha )\}$$\mathrm{η}(K):=inf\left\{H_K(\mathrm{Î\pm }):K=\mathbb{Q}(\mathrm{Î\pm })\right\}$eta(K):=i n f{H_(K)(alpha):K=Q(alpha)}\eta(K):=\inf \left\{H_{K}(\alpha): K=\mathbb{Q}(\alpha)\right\}$\mathrm{η}(K):=inf\{H_K(\mathrm{Î\pm }):K=\mathbb{Q}(\mathrm{Î\pm })\}$ denote the minimum (relative) multiplicative Weil height of a generating element of $K$$K$KK$K$. Roughly speaking, Ellenberg notes the criterion (2.4) can actually allow prime ideals with norms as large as $\eta (K{)}^{1/\ell }$$\mathrm{η}(K{)}^{1/\mathrm{â„“}}$eta(K)^(1//â„“)\eta(K)^{1 / \ell}$\mathrm{η}(K{)}^{1/\mathrm{â„“}}$. The restriction to norms $<D_K^{\frac{1}{2\ell (n-1)}}$$<D_K^{\frac{1}{2\mathrm{â„“}(n−1)}}$< D_(K)^((1)/(2â„“(n-1)))<D_{K}^{\frac{1}{2 \ell(n-1)}}$<D_K^{\frac{1}{2\mathrm{â„“}(n−1)}}$ in (2.4) was made since the lower bound $\eta (K)\ge D_K^{\frac{1}{2(n-1)}}$$\mathrm{η}(K)â‰\yen D_K^{\frac{1}{2(n−1)}}$eta(K) >= D_(K)^((1)/(2(n-1)))\eta(K) \geq D_{K}^{\frac{1}{2(n-1)}}$\mathrm{η}(K)â‰\yen D_K^{\frac{1}{2(n−1)}}$ holds for all fields [80]. Widmer, also with Frei, has shown that $\eta (K)$$\mathrm{η}(K)$eta(K)\eta(K)$\mathrm{η}(K)$ can be enlarged for almost all fields in certain families, leading to improved upper bounds for $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion in those fields [41,95]. That is, they improve the very notion of the "GRH-bound" (2.5), and show that the parameter we have called ${\mathrm{\Delta }}_{\text{GRH}}$${\mathrm{Δ}}_{\text{GRH }}$Delta_("GRH ")\Delta_{\text {GRH }}${\mathrm{Δ}}_{\text{GRH }}$ can actually be taken smaller for some fields. Their work raises interesting open questions: what upper and lower bounds hold for $\eta (K)$$\mathrm{η}(K)$eta(K)\eta(K)$\mathrm{η}(K)$, for all (or almost all) fields in a family? Ruppert [74] has conjectured uniform upper bounds $\eta (K)\le D_K^{1/2}$$\mathrm{η}(K)≤D_K^{1/2}$eta(K) <= D_(K)^(1//2)\eta(K) \leq D_{K}^{1 / 2}$\mathrm{η}(K)≤D_K^{1/2}$ (now proved for almost all fields in some families by [72]). If this is true, the Ellenberg-Venkatesh criterion would hit a barrier, for most fields, with a result like $|{\mathrm{C}\mathrm{l}}_K[\ell ]|\ll D_K^{1/2-1/2\ell +\epsilon }$$\left|{\mathrm{C}\mathrm{l}}_K[\mathrm{â„“}]\right|≪D_K^{1/2−1/2\mathrm{â„“}+\mathrm{Î\mu }}$|Cl_(K)[â„“]|≪D_(K)^(1//2-1//2â„“+epsi)\left|\mathrm{Cl}_{K}[\ell]\right| \ll D_{K}^{1 / 2-1 / 2 \ell+\varepsilon}$|{\mathrm{C}\mathrm{l}}_K[\mathrm{â„“}]|≪D_K^{1/2−1/2\mathrm{â„“}+\mathrm{Î\mu }}$ for any degree $n$$n$nn$n$, still far from the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture. It would be very interesting to find a new, different criterion.

Recall that the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture 2.1 is still known only in the case stemming from Gauss's work, namely for $n=2,\ell =2$$n=2,\mathrm{â„“}=2$n=2,â„“=2n=2, \ell=2$n=2,\mathrm{â„“}=2$. It is a good idea to affirm why we believe the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture should be true. We will consider this from three perspectives.

So far, when we have mentioned a result for almost all fields in a family, we have not focused on the size of a potential exceptional set, other than showing it is smaller than the size of the full family. But to understand the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture, we must quantify a potential exceptional set, and show that for all sufficiently large discriminants, it is empty.

Let us abstract this, for a family ${\mathcal{F}}_0(X)$${\mathcal{F}}_0(X)$F_(0)(X)\mathscr{F}_{0}(X)${\mathcal{F}}_0(X)$ of fields $K$$K$KK$K$ with $D_K$$D_K$D_(K)D_{K}$D_K$ in a dyadic range $(X/2,X]$$(X/2,X]$(X//2,X](X / 2, X]$(X/2,X]$, from which more general results can easily be deduced by summing over $\ll \log X$$≪\log ⁡X$≪log X\ll \log X$≪\log ⁡X$ dyadic ranges. Suppose $f:{\mathcal{F}}_0(X)\to \mathbb{N}$$f:{\mathcal{F}}_0(X)→\mathbb{N}$f:F_(0)(X)rarrNf: \mathscr{F}_{0}(X) \rightarrow \mathbb{N}$f:{\mathcal{F}}_0(X)→\mathbb{N}$ is a function with $f(K)\le D_K^a$$f(K)≤D_K^a$f(K) <= D_(K)^(a)f(K) \leq D_{K}^{a}$f(K)≤D_K^a$ for all $K$$K$KK$K$. Suppose that for some $\mathrm{\Delta }<a$$\mathrm{Δ}<a$Delta < a\Delta<a$\mathrm{Δ}<a$ we can improve this to $f(K)\le D_K^{\mathrm{\Delta }}$$f(K)≤D_K^{\mathrm{Δ}}$f(K) <= D_(K)^(Delta)f(K) \leq D_{K}^{\Delta}$f(K)≤D_K^{\mathrm{Δ}}$ for all $K$$K$KK$K$ outside of some exceptional set $E_0^{\mathrm{\Delta }}(X)\subset {\mathcal{F}}_0(X)$$E_0^{\mathrm{Δ}}(X)⊂{\mathcal{F}}_0(X)$E_(0)^(Delta)(X)subF_(0)(X)E_{0}^{\Delta}(X) \subset \mathscr{F}_{0}(X)$E_0^{\mathrm{Δ}}(X)⊂{\mathcal{F}}_0(X)$. Then

$\sum _{K\in {\mathcal{F}}_0(X)}f(K)=\sum _{K\in {\mathcal{F}}_0(X)\mathrm{\setminus }{E}_0^{\mathrm{\Delta }}(X)}f(K)+\sum _{K\in E_0^{\mathrm{\Delta }}(X)}f(K)\le |{\mathcal{F}}_0(X)|X^{\mathrm{\Delta }}+|E_0^{\mathrm{\Delta }}(X)|X^a$$\underset{K∈{\mathcal{F}}_0(X)}{∑} f(K)=\underset{K∈{\mathcal{F}}_0(X)\mathrm{∖}{E}_0^{\mathrm{Δ}}(X)}{∑} f(K)+\underset{K∈E_0^{\mathrm{Δ}}(X)}{∑} f(K)≤\left|{\mathcal{F}}_0(X)\right|X^{\mathrm{Δ}}+\left|E_0^{\mathrm{Δ}}(X)\right|X^a$sum_(K inF_(0)(X))f(K)=sum_(K inF_(0)(X)\\E_(0)^(Delta)(X))f(K)+sum_(K inE_(0)^(Delta)(X))f(K) <= |F_(0)(X)|X^(Delta)+|E_(0)^(Delta)(X)|X^(a)\sum_{K \in \mathscr{F}_{0}(X)} f(K)=\sum_{K \in \mathscr{F}_{0}(X) \backslash E_{0}^{\Delta}(X)} f(K)+\sum_{K \in E_{0}^{\Delta}(X)} f(K) \leq\left|\mathscr{F}_{0}(X)\right| X^{\Delta}+\left|E_{0}^{\Delta}(X)\right| X^{a}$\underset{K∈{\mathcal{F}}_0(X)}{∑}f(K)=\underset{K∈{\mathcal{F}}_0(X)\mathrm{∖}{E}_0^{\mathrm{Δ}}(X)}{∑}f(K)+\underset{K∈E_0^{\mathrm{Δ}}(X)}{∑}f(K)≤|{\mathcal{F}}_0(X)|X^{\mathrm{Δ}}+|E_0^{\mathrm{Δ}}(X)|X^a$.

As long as $|E_0^{\mathrm{\Delta }}(X)|\ll |{\mathcal{F}}_0(X)|X^{-(a-\mathrm{\Delta })}$$\left|E_0^{\mathrm{Δ}}(X)\right|≪\left|{\mathcal{F}}_0(X)\right|X^{−(a−\mathrm{Δ})}$|E_(0)^(Delta)(X)|≪|F_(0)(X)|X^(-(a-Delta))\left|E_{0}^{\Delta}(X)\right| \ll\left|\mathscr{F}_{0}(X)\right| X^{-(a-\Delta)}$|E_0^{\mathrm{Δ}}(X)|≪|{\mathcal{F}}_0(X)|X^{−(a−\mathrm{Δ})}$, this shows that $f(K)\ll X^{\mathrm{\Delta }}$$f(K)≪X^{\mathrm{Δ}}$f(K)≪X^(Delta)f(K) \ll X^{\Delta}$f(K)≪X^{\mathrm{Δ}}$ on average. On the other hand, suppose we know $\sum _{K\in {\mathcal{F}}_0(X)}f(K)\le X^b$$\underset{K∈{\mathcal{F}}_0(X)}{∑} f(K)≤X^b$sum_(K inF_(0)(X))f(K) <= X^(b)\sum_{K \in \mathscr{F}_{0}(X)} f(K) \leq X^{b}$\underset{K∈{\mathcal{F}}_0(X)}{∑}f(K)≤X^b$. Then a potential set of exceptions $E_0^{\mathrm{\Delta }}(X)=\{K\in {\mathcal{F}}_0(X):f(K)>D_K^{\mathrm{\Delta }}\}$$E_0^{\mathrm{Δ}}(X)=\left\{K∈{\mathcal{F}}_0(X):f(K)>D_K^{\mathrm{Δ}}\right\}$E_(0)^(Delta)(X)={K inF_(0)(X):f(K) > D_(K)^(Delta)}E_{0}^{\Delta}(X)=\left\{K \in \mathscr{F}_{0}(X): f(K)>D_{K}^{\Delta}\right\}$E_0^{\mathrm{Δ}}(X)=\{K∈{\mathcal{F}}_0(X):f(K)>D_K^{\mathrm{Δ}}\}$ can be controlled by

Thus $|E_0^{\mathrm{\Delta }}(X)|\ll X^{b-\mathrm{\Delta }}$$\left|E_0^{\mathrm{Δ}}(X)\right|≪X^{b−\mathrm{Δ}}$|E_(0)^(Delta)(X)|≪X^(b-Delta)\left|E_{0}^{\Delta}(X)\right| \ll X^{b-\Delta}$|E_0^{\mathrm{Δ}}(X)|≪X^{b−\mathrm{Δ}}$, and exceptional fields are density zero in ${\mathcal{F}}_0(X)$${\mathcal{F}}_0(X)$F_(0)(X)\mathscr{F}_{0}(X)${\mathcal{F}}_0(X)$, provided $X^{b-\mathrm{\Delta }}=o\left(|{\mathcal{F}}_0(X)|\right)$$X^{b−\mathrm{Δ}}=o\left(\left|{\mathcal{F}}_0(X)\right|\right)$X^(b-Delta)=o(|F_(0)(X)|)X^{b-\Delta}=o\left(\left|\mathscr{F}_{0}(X)\right|\right)$X^{b−\mathrm{Δ}}=o\left(|{\mathcal{F}}_0(X)|\right)$. That is, a nontrivial upper bound on $\ell$$\mathrm{ℓ}$ℓ\ell$\mathrm{ℓ}$-torsion for "almost all" fields in a family $\mathcal{F}$$\mathcal{F}$F\mathscr{F}$\mathcal{F}$ is essentially equivalent to the same upper bound "on average."

To verify the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture, we wish to show a "pointwise" bound: for every $\epsilon >0$$\mathrm{Î\mu }>0$epsi > 0\varepsilon>0$\mathrm{Î\mu }>0$, there exists $D_{\epsilon }$$D_{\mathrm{Î\mu }}$D_(epsi)D_{\varepsilon}$D_{\mathrm{Î\mu }}$ such that when $D_K\ge D_{\epsilon }$$D_Kâ‰\yen D_{\mathrm{Î\mu }}$D_(K) >= D_(epsi)D_{K} \geq D_{\varepsilon}$D_Kâ‰\yen D_{\mathrm{Î\mu }}$, there are no exceptions to the bound $|{\mathrm{C}\mathrm{l}}_K[\ell ]|\le D_K^{\epsilon }$$\left|{\mathrm{C}\mathrm{l}}_K[\mathrm{â„“}]\right|≤D_K^{\mathrm{Î\mu }}$|Cl_(K)[â„“]| <= D_(K)^(epsi)\left|\mathrm{Cl}_{K}[\ell]\right| \leq D_{K}^{\varepsilon}$|{\mathrm{C}\mathrm{l}}_K[\mathrm{â„“}]|≤D_K^{\mathrm{Î\mu }}$. The key is to consider not averages but arbitrarily high $k$$k$kk$k$ th moments. In the general setting above, suppose that we know $\sum _{K\in {\mathcal{F}}_0(X)}f(K{)}^k\le X^b$$\underset{K∈{\mathcal{F}}_0(X)}{∑} f(K{)}^k≤X^b$sum_(K inF_(0)(X))f(K)^(k) <= X^(b)\sum_{K \in \mathscr{F}_{0}(X)} f(K)^{k} \leq X^{b}$\underset{K∈{\mathcal{F}}_0(X)}{∑}f(K{)}^k≤X^b$, for a real number $k\ge 1$$kâ‰\yen 1$k >= 1k \geq 1$kâ‰\yen 1$. Then for any fixed $\mathrm{\Delta }>0$$\mathrm{Δ}>0$Delta > 0\Delta>0$\mathrm{Δ}>0$, adapting the argument (5.2) shows that $|E_0^{\mathrm{\Delta }}(X)|\ll X^{b-k\mathrm{\Delta }}$$\left|E_0^{\mathrm{Δ}}(X)\right|≪X^{b−k\mathrm{Δ}}$|E_(0)^(Delta)(X)|≪X^(b-k Delta)\left|E_{0}^{\Delta}(X)\right| \ll X^{b-k \Delta}$|E_0^{\mathrm{Δ}}(X)|≪X^{b−k\mathrm{Δ}}$. If the $k$$k$kk$k$ th moment is uniformly bounded by $X^b$$X^b$X^(b)X^{b}$X^b$ for a sequence of $k\to \mathrm{\infty }$$k→\mathrm{∞}$k rarr ook \rightarrow \infty$k→\mathrm{∞}$, then for each $\mathrm{\Delta }>0$$\mathrm{Δ}>0$Delta > 0\Delta>0$\mathrm{Δ}>0$, we can take $k$$k$kk$k$ sufficiently large to conclude that the set of exceptions is empty.

This perspective has been applied by Pierce, Turnage-Butterbaugh, and Wood in [73] to prove that the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture holds for all fields in a family $\mathcal{F}(X)$$\mathcal{F}(X)$F(X)\mathscr{F}(X)$\mathcal{F}(X)$ if there is a real number $\alpha \ge 1$$\mathrm{Î\pm }â‰\yen 1$alpha >= 1\alpha \geq 1$\mathrm{Î\pm }â‰\yen 1$ such that for a sequence of arbitrarily large $k$$k$kk$k$,

The Cohen-Lenstra-Martinet heuristics predict that (5.3) holds, in the form of an even stronger asymptotic with $\alpha =1$$\mathrm{Î\pm }=1$alpha=1\alpha=1$\mathrm{Î\pm }=1$, for all integers $k\ge 1$$kâ‰\yen 1$k >= 1k \geq 1$kâ‰\yen 1$, for families of Galois $G$$G$GG$G$-extensions, at least for all primes $\ell \nmid |G|$$\mathrm{â„“}∤|G|$ℓ∤|G|\ell \nmid|G|$\mathrm{â„“}∤|G|$. The appropriate moment formulation can be found in [21] for degree 2 fields and in [92] for higher degrees, building on [22]. This confirms that the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture follows from another well-known set of conjectures.

The Cohen-Lenstra-Martinet heuristics are a subject of intense interest and much recent activity. Here are some spectacular successes most closely related to our topic. Dav-
enport and Heilbronn [28] have proved

second-order terms have been found in [5,11,85]. Bhargava [7] has proved

in which each isomorphism class of fields is counted once. Very recently, [63] obtained analogues of (5.4) for averages over ${\mathcal{F}}_{2^m}(G;X)$${\mathcal{F}}_{2^m}(G;X)$F_(2^(m))(G;X)\mathscr{F}_{2^{m}}(G ; X)${\mathcal{F}}_{2^m}(G;X)$ for any permutation group $G\subset S_{2^m}$$G⊂S_{2^m}$G subS_(2^(m))G \subset S_{2^{m}}$G⊂S_{2^m}$ that is a transitive permutation 2-group containing a transposition. See also the work of Smith on the distribution of $2^k$$2^k$2^(k)2^{k}$2^k$-class groups in imaginary quadratic fields [81]; Koymans and Pagano on ${\ell }^k$${\mathrm{â„“}}^k$â„“^(k)\ell^{k}${\mathrm{â„“}}^k$-class groups of degree $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$ cyclic fields [59]; Klys on moments of $p$$p$pp$p$-torsion in cyclic degree $p$$p$pp$p$ fields (conditional on GRH for $p\ge 5$$pâ‰\yen 5$p >= 5p \geq 5$pâ‰\yen 5$ ) [55]; Milovic and Koymans on 16-rank in quadratic fields [57,58]; Bhargava and Varma [13,14] elaborating on (5.4) and (5.5).

The perspective of moments (5.3) provides a strong motivation to prove the $k$$k$kk$k$ th moment bounds for $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion. Fouvry and Klüners have proved an asymptotic for the $k$$k$kk$k$ th moments related to 4-torsion when $K$$K$KK$K$ is quadratic, for all integers $k\ge 1$$kâ‰\yen 1$k >= 1k \geq 1$kâ‰\yen 1$ [38]. Heath-Brown and Pierce have proved nontrivial bounds for the $k$$k$kk$k$ th moments of $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion for imaginary quadratic fields, for all odd primes $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$ [46]. For example, they establish second moment bounds

as well as results for the $k$$k$kk$k$ th moments for all $k\ge 1$$kâ‰\yen 1$k >= 1k \geq 1$kâ‰\yen 1$. In general, proving tighter control on the size of an exceptional family $E_0^{\mathrm{\Delta }}(X)$$E_0^{\mathrm{Δ}}(X)$E_(0)^(Delta)(X)E_{0}^{\Delta}(X)$E_0^{\mathrm{Δ}}(X)$ can be used to deduce a better moment bound for $|{\mathrm{C}\mathrm{l}}_K[\ell ]|$$\left|{\mathrm{C}\mathrm{l}}_K[\mathrm{â„“}]\right|$|Cl_(K)[â„“]|\left|\mathrm{Cl}_{K}[\ell]\right|$|{\mathrm{C}\mathrm{l}}_K[\mathrm{â„“}]|$, similar to (5.1). This has recently been exploited by Frei and Widmer, in combination with refinements of the Ellenberg-Venkatesh criterion, to improve moment bounds on $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion for the families of fields studied in [72] (if $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$ is sufficiently large); see [41].

Let us mention a connection to elliptic curves; this was after all the setting in which Brumer and Silverman initially posed the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture. Let $E(q)$$E(q)$E(q)E(q)$E(q)$ denote the number of isomorphism classes of elliptic curves over $\mathbb{Q}$$\mathbb{Q}$Q\mathbb{Q}$\mathbb{Q}$ with conductor $q$$q$qq$q$. Brumer and Silverman have conjectured that $E(q){\ll }_{\epsilon }{q}^{\epsilon }$$E(q){≪}_{\mathrm{Î\mu }}{q}^{\mathrm{Î\mu }}$E(q)≪_(epsi)q^(epsi)E(q) \ll_{\varepsilon} q^{\varepsilon}$E(q){≪}_{\mathrm{Î\mu }}{q}^{\mathrm{Î\mu }}$ for every $q\ge 1,\epsilon >0$$qâ‰\yen 1,\mathrm{Î\mu }>0$q >= 1,epsi > 0q \geq 1, \varepsilon>0$qâ‰\yen 1,\mathrm{Î\mu }>0$ [17]. Conditionally, this follows from GRH combined with a weak form of the Birch-Swinnerton-Dyer conjecture. They also showed this follows from the 3-torsion Conjecture for quadratic fields, by proving

Duke and Kowalski have combined this with the celebrated asymptotic (5.4) to bound $\sum _{1\le q\le Q}E(q)\ll Q^{1+\epsilon }$$\underset{1≤q≤Q}{∑} E(q)≪Q^{1+\mathrm{Î\mu }}$sum_(1 <= q <= Q)E(q)≪Q^(1+epsi)\sum_{1 \leq q \leq Q} E(q) \ll Q^{1+\varepsilon}$\underset{1≤q≤Q}{∑}E(q)≪Q^{1+\mathrm{Î\mu }}$ for every $\epsilon >0$$\mathrm{Î\mu }>0$epsi > 0\varepsilon>0$\mathrm{Î\mu }>0$ [30]. (See also [39] for ordering by discriminant.) Pierce, Turnage-Butterbaugh, and Wood have recently proved that for all $k\ge 1$$kâ‰\yen 1$k >= 1k \geq 1$kâ‰\yen 1$, the $k$$k$kk$k$ th moment of 3-torsion in quadratic fields dominates the $\gamma k$$\mathrm{γ}k$gamma k\gamma k$\mathrm{γ}k$ th moment of $E(q)$$E(q)$E(q)E(q)$E(q)$, for a numerical constant $\gamma \approx 1.9745\dots$$\mathrm{γ}≈1.9745…$gamma~~1.9745 dots\gamma \approx 1.9745 \ldots$\mathrm{γ}≈1.9745…$ coming from [48], which sharpened the relation (5.7). Thus
new moment bounds for $E(q)$$E(q)$E(q)E(q)$E(q)$ can be obtained from (5.6), for example. Here is an open problem: prove that $\sum _{1\le q\le Q}E(q)=o(Q)$$\underset{1≤q≤Q}{∑} E(q)=o(Q)$sum_(1 <= q <= Q)E(q)=o(Q)\sum_{1 \leq q \leq Q} E(q)=o(Q)$\underset{1≤q≤Q}{∑}E(q)=o(Q)$. This would show for the first time that integers that are the conductor of an elliptic curve have density zero in $\mathbb{Z}$$\mathbb{Z}$Z\mathbb{Z}$\mathbb{Z}$. In fact, it is conjectured by Watkins that this average is asymptotic to $c{Q}^{5/6}$$c{Q}^{5/6}$cQ^(5//6)c Q^{5 / 6}$c{Q}^{5/6}$ for a certain constant $c$$c$cc$c$ [94] (building on an analogous conjecture by Brumer-McGuinness for ordering by discriminant [16]).

To conclude, in this section we saw that the truth of the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture is implied by the truth of the well-known Cohen-Lenstra-Martinet heuristics on the distribution of class groups.

Let $K/\mathbb{Q}$$K/\mathbb{Q}$K//QK / \mathbb{Q}$K/\mathbb{Q}$ be a degree $n$$n$nn$n$ extension. The Hilbert class field $H_K$$H_K$H_(K)H_{K}$H_K$ is the maximal abelian unramified extension of $K$$K$KK$K$, and ${\mathrm{C}\mathrm{l}}_K$${\mathrm{C}\mathrm{l}}_K$Cl_(K)\mathrm{Cl}_{K}${\mathrm{C}\mathrm{l}}_K$ is isomorphic to $\mathrm{Gal}\left(H_K/K\right)$$\mathrm{Gal}⁡\left(H_K/K\right)$Gal(H_(K)//K)\operatorname{Gal}\left(H_{K} / K\right)$\mathrm{Gal}⁡\left(H_K/K\right)$. A second way to motivate the $\ell$$\mathrm{ℓ}$ℓ\ell$\mathrm{ℓ}$-torsion Conjecture is to count intermediate fields between $K$$K$KK$K$ and $H_K$$H_K$H_(K)H_{K}$H_K$.

Here is an argument recorded by Pierce, Turnage-Butterbaugh, and Wood in [73]. Fix a prime $\ell$$\mathrm{ℓ}$ℓ\ell$\mathrm{ℓ}$ and write ${\mathrm{C}\mathrm{l}}_K$${\mathrm{C}\mathrm{l}}_K$Cl_(K)\mathrm{Cl}_{K}${\mathrm{C}\mathrm{l}}_K$ additively, so that ${\mathrm{C}\mathrm{l}}_K[\ell ]\simeq {\mathrm{C}\mathrm{l}}_K/\ell {\mathrm{C}\mathrm{l}}_K$${\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]≃{\mathrm{C}\mathrm{l}}_K/\mathrm{ℓ}{\mathrm{C}\mathrm{l}}_K$Cl_(K)[ℓ]≃Cl_(K)//ℓCl_(K)\mathrm{Cl}_{K}[\ell] \simeq \mathrm{Cl}_{K} / \ell \mathrm{Cl}_{K}${\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]≃{\mathrm{C}\mathrm{l}}_K/\mathrm{ℓ}{\mathrm{C}\mathrm{l}}_K$. Now define the fixed field $L=H_K^{\ell {\mathrm{C}\mathrm{l}}_K}$$L=H_K^{\mathrm{ℓ}{\mathrm{C}\mathrm{l}}_K}$L=H_(K)^(ℓCl_(K))L=H_{K}^{\ell \mathrm{Cl}_{K}}$L=H_K^{\mathrm{ℓ}{\mathrm{C}\mathrm{l}}_K}$ lying between $K$$K$KK$K$ and $H_K$$H_K$H_(K)H_{K}$H_K$, so $\mathrm{Gal}(L/K)\simeq {\mathrm{C}\mathrm{l}}_K[\ell ]$$\mathrm{Gal}⁡(L/K)≃{\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]$Gal(L//K)≃Cl_(K)[ℓ]\operatorname{Gal}(L / K) \simeq \mathrm{Cl}_{K}[\ell]$\mathrm{Gal}⁡(L/K)≃{\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]$. Each surjection ${\mathrm{C}\mathrm{l}}_K[\ell ]\to \mathbb{Z}/\ell \mathbb{Z}$${\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]→\mathbb{Z}/\mathrm{ℓ}\mathbb{Z}$Cl_(K)[ℓ]rarrZ//ℓZ\mathrm{Cl}_{K}[\ell] \rightarrow \mathbb{Z} / \ell \mathbb{Z}${\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]→\mathbb{Z}/\mathrm{ℓ}\mathbb{Z}$ generates an intermediate field $M$$M$MM$M$, with $K\subset M\subset L$$K⊂M⊂L$K sub M sub LK \subset M \subset L$K⊂M⊂L$ and $\mathrm{deg}(M/\mathbb{Q})=n\ell$$\mathrm{deg}⁡(M/\mathbb{Q})=n\mathrm{ℓ}$deg(M//Q)=nℓ\operatorname{deg}(M / \mathbb{Q})=n \ell$\mathrm{deg}⁡(M/\mathbb{Q})=n\mathrm{ℓ}$. If $|{\mathrm{C}\mathrm{l}}_K[\ell ]|={\ell }^r$$\left|{\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]\right|={\mathrm{ℓ}}^r$|Cl_(K)[ℓ]|=ℓ^(r)\left|\mathrm{Cl}_{K}[\ell]\right|=\ell^{r}$|{\mathrm{C}\mathrm{l}}_K[\mathrm{ℓ}]|={\mathrm{ℓ}}^r$, say, this produces $\approx {\ell }^{r-1}$$≈{\mathrm{ℓ}}^{r−1}$~~ℓ^(r-1)\approx \ell^{r-1}$≈{\mathrm{ℓ}}^{r−1}$ such fields $M$$M$MM$M$. The crucial point is that since $H_K$$H_K$H_(K)H_{K}$H_K$ is an unramified extension, all these fields satisfy a rigid discriminant identity $D_M=D_K^{\ell }$$D_M=D_K^{\mathrm{ℓ}}$D_(M)=D_(K)^(ℓ)D_{M}=D_{K}^{\ell}$D_M=D_K^{\mathrm{ℓ}}$. Consequently, if we can count how many number fields of degree $n\ell$$n\mathrm{ℓ}$nℓn \ell$n\mathrm{ℓ}$ can share the same fixed discriminant, then we can bound $\ell$$\mathrm{ℓ}$ℓ\ell$\mathrm{ℓ}$-torsion in ${\mathrm{C}\mathrm{l}}_K$${\mathrm{C}\mathrm{l}}_K$Cl_(K)\mathrm{Cl}_{K}${\mathrm{C}\mathrm{l}}_K$. (We have seen this problem before.) We formalize the problem of counting number fields of fixed discriminant as follows:

Property ${\mathbf{D}}_n(\mathrm{\Delta })$${\mathbf{D}}_n(\mathrm{Δ})$D_(n)(Delta)\mathbf{D}_{n}(\Delta)${\mathbf{D}}_n(\mathrm{Δ})$. Fix a degree $n\ge 2$$nâ‰\yen 2$n >= 2n \geq 2$nâ‰\yen 2$. Property ${\mathbf{D}}_n(\mathrm{\Delta })$${\mathbf{D}}_n(\mathrm{Δ})$D_(n)(Delta)\mathbf{D}_{n}(\Delta)${\mathbf{D}}_n(\mathrm{Δ})$ holds if for every $\epsilon >0$$\mathrm{Î\mu }>0$epsi > 0\varepsilon>0$\mathrm{Î\mu }>0$ and for every fixed integer $D>1$$D>1$D > 1D>1$D>1$, at most ${\ll }_{n,\epsilon }{D}^{\mathrm{\Delta }+\epsilon }$${≪}_{n,\mathrm{Î\mu }}{D}^{\mathrm{Δ}+\mathrm{Î\mu }}$≪_(n,epsi)D^(Delta+epsi)\ll_{n, \varepsilon} D^{\Delta+\varepsilon}${≪}_{n,\mathrm{Î\mu }}{D}^{\mathrm{Δ}+\mathrm{Î\mu }}$ fields $K/\mathbb{Q}$$K/\mathbb{Q}$K//QK / \mathbb{Q}$K/\mathbb{Q}$ of degree $n$$n$nn$n$ have $D_K=D$$D_K=D$D_(K)=DD_{K}=D$D_K=D$.

The strategy sketched above ultimately proves that property ${\mathbf{D}}_{n\ell }(\mathrm{\Delta })$${\mathbf{D}}_{n\mathrm{ℓ}}(\mathrm{Δ})$D_(nℓ)(Delta)\mathbf{D}_{n \ell}(\Delta)${\mathbf{D}}_{n\mathrm{ℓ}}(\mathrm{Δ})$ implies ${\mathbf{C}}_{n,\ell }(\ell \mathrm{\Delta })$${\mathbf{C}}_{n,\mathrm{ℓ}}(\mathrm{ℓ}\mathrm{Δ})$C_(n,ℓ)(ℓDelta)\mathbf{C}_{n, \ell}(\ell \Delta)${\mathbf{C}}_{n,\mathrm{ℓ}}(\mathrm{ℓ}\mathrm{Δ})$. This leads inevitably to the question: is property ${\mathbf{D}}_{n\ell }(0)$${\mathbf{D}}_{n\mathrm{ℓ}}(0)$D_(nℓ)(0)\mathbf{D}_{n \ell}(0)${\mathbf{D}}_{n\mathrm{ℓ}}(0)$ true? Here is a conjecture:

Conjecture 5.1 (Discriminant multiplicity conjecture). For each $n\ge 2$$nâ‰\yen 2$n >= 2n \geq 2$nâ‰\yen 2$, for every $\epsilon >0$$\mathrm{Î\mu }>0$epsi > 0\varepsilon>0$\mathrm{Î\mu }>0$, and for every integer $D>1$$D>1$D > 1D>1$D>1$, at most ${\ll }_{n,\epsilon }{D}^{\epsilon }$${≪}_{n,\mathrm{Î\mu }}{D}^{\mathrm{Î\mu }}$≪_(n,epsi)D^(epsi)\ll_{n, \varepsilon} D^{\varepsilon}${≪}_{n,\mathrm{Î\mu }}{D}^{\mathrm{Î\mu }}$ fields $K/\mathbb{Q}$$K/\mathbb{Q}$K//QK / \mathbb{Q}$K/\mathbb{Q}$ of degree $n$$n$nn$n$ have $D_K=D$$D_K=D$D_(K)=DD_{K}=D$D_K=D$.

This conjecture has been recorded by Duke [29]. It implies the $\ell$$\mathrm{â„“}$â„“\ell$\mathrm{â„“}$-torsion Conjecture, a link noted in $[29,35]$$[29,35]$[29,35][29,35]$[29,35]$ and quantified in [73]. Recall the conjecture (3.3) for counting all fields of degree $n$$n$nn$n$ and discriminant $D_K\le X$$D_K≤X$D_(K) <= XD_{K} \leq X$D_K≤X$. The Discriminant Multiplicity Conjecture for degree $n$$n$nn$n$ would immediately imply $N_n(X)\ll X^{1+\epsilon }$$N_n(X)≪X^{1+\mathrm{Î\mu }}$N_(n)(X)≪X^(1+epsi)N_{n}(X) \ll X^{1+\varepsilon}$N_n(X)≪X^{1+\mathrm{Î\mu }}$, which indicates its level of difficulty. Of course, in general, property ${\mathbf{D}}_n(\mathrm{\Delta })$${\mathbf{D}}_n(\mathrm{Δ})$D_(n)(Delta)\mathbf{D}_{n}(\Delta)${\mathbf{D}}_n(\mathrm{Δ})$ implies $N_n(X)\ll X^{1+\mathrm{\Delta }+\epsilon }$$N_n(X)≪X^{1+\mathrm{Δ}+\mathrm{Î\mu }}$N_(n)(X)≪X^(1+Delta+epsi)N_{n}(X) \ll X^{1+\Delta+\varepsilon}$N_n(X)≪X^{1+\mathrm{Δ}+\mathrm{Î\mu }}$ for all $\epsilon >0$$\mathrm{Î\mu }>0$epsi > 0\varepsilon>0$\mathrm{Î\mu }>0$. (In terms of lower bounds, Ellenberg and Venkatesh have noted there can be $\gg D^{c/\log \log D}$$≫D^{c/\log ⁡\log ⁡D}$≫D^(c//log log D)\gg D^{c / \log \log D}$≫D^{c/\log ⁡\log ⁡D}$ extensions $K/\mathbb{Q}$$K/\mathbb{Q}$K//QK / \mathbb{Q}$K/\mathbb{Q}$ with a fixed Galois group and fixed discriminant $D$$D$DD$D$ [35].)

The Discriminant Multiplicity Conjecture posits that ${\mathbf{D}}_n(0)$${\mathbf{D}}_n(0)$D_(n)(0)\mathbf{D}_{n}(0)${\mathbf{D}}_n(0)$ holds for each $n\ge 2$$nâ‰\yen 2$n >= 2n \geq 2$nâ‰\yen 2$. This is true for $n=2$$n=2$n=2n=2$n=2$, but it is not known for any other degree. For degrees $n=3,4,5$$n=3,4,5$n=3,4,5n=3,4,5$n=3,4,5$, the best-known results currently are ${\mathbf{D}}_3(1/3)$${\mathbf{D}}_3(1/3)$D_(3)(1//3)\mathbf{D}_{3}(1 / 3)${\mathbf{D}}_3(1/3)$ by [34]; ${\mathbf{D}}_4(1/2)$${\mathbf{D}}_4(1/2)$D_(4)(1//2)\mathbf{D}_{4}(1 / 2)${\mathbf{D}}_4(1/2)$ as found in [52, 72, 73, 97]; ${\mathbf{D}}_5(199/200)$${\mathbf{D}}_5(199/200)$D_(5)(199//200)\mathbf{D}_{5}(199 / 200)${\mathbf{D}}_5(199/200)$ as found in [33], building on [9,79]. Currently for $n\ge 6$$nâ‰\yen 6$n >= 6n \geq 6$nâ‰\yen 6$, the only result for
${\mathbf{D}}_n(\mathrm{\Delta })$${\mathbf{D}}_n(\mathrm{Δ})$D_(n)(Delta)\mathbf{D}_{n}(\Delta)${\mathbf{D}}_n(\mathrm{Δ})$ is a trivial consequence of counting fields of bounded discriminant, as in (3.4), so in particular $\mathrm{\Delta }=c_0(\log n{)}^2>1$$\mathrm{Δ}=c_0(\log ⁡n{)}^2>1$Delta=c_(0)(log n)^(2) > 1\Delta=c_{0}(\log n)^{2}>1$\mathrm{Δ}=c_0(\log ⁡n{)}^2>1$ in those cases. It would be very interesting to improve the exponent known for ${\mathbf{D}}_n(\mathrm{\Delta })$${\mathbf{D}}_n(\mathrm{Δ})$D_(n)(Delta)\mathbf{D}_{n}(\Delta)${\mathbf{D}}_n(\mathrm{Δ})$, for any fixed degree $n\ge 3$$nâ‰\yen 3$n >= 3n \geq 3$nâ‰\yen 3$.